## “Cannot be determined by the information given”

Michael drives x miles due north at arrives at Point A. He then heads due east for y miles. Finally, he drives z miles in a straight line until he reaches his starting point. If x, y, and z are integers, then how many miles did Michael drive if the shortest leg was 5 miles?

(A) 5 miles

(B) 12 miles

(C) 25 miles

(D) 30 miles

(E) Cannot be determined by the information given.

Historically, answer choice (E), with that exact wording, “cannot be determined…”, is usually a trap. If your initial response to the question is to throw your hands in the air exclaiming, “I don’t know!” answer choice (E) is most likely a trap.

(One important thing of note: the answer choice “(E) cannot be…” should not be mixed up with Quantitative Comparison answer choice (D).)

I’m guessing with this problem that may very well have been your response. If so, take another look at the problem and see if you can crack it. And don’t listen to that voice – oftentimes the seemingly insolvable is solvable. By giving up and choosing (E), you are falling for the GRE’s nefarious schemes. (In this case I’m the nefarious one, as I wrote the problem).

Hopefully, you took a minute—so let’s take the problem apart. If you drew the problem out – as you should with any “driving problem” (it usually entails somebody driving due north, etc.), then you will notice that Michael’s path describes a triangle.

If you are still thinking the problem is unsolvable, remember that the GRE will hide important information in the middle of the sentence. In this case, the fact that x,y, and z are both integers is crucial. Remember the triangle – when north meets east a right angle is formed. If the shortest side is 5 miles, and all the sides are integers, there is only one triangle that fits the bill: the 5:12:13 triangle.

The question is asking for the number of miles Mike drove. Therefore, we have to add up the perimeter of a 5:12:13 triangle, which gives us 30. (D).

## A Little Twist

Now that you’ve got the hang of it, let’s try the exact same problem. However, this time around I am going to change something. See if you can figure out what it is, and then see if you can answer the question correctly.

Michael drives x miles due north at arrives at Point A. He then heads due east for y miles. Finally, he drives z miles a straight line towards is starting point. If x, y, and z are integers, then how many miles did Michael drive if one of the legs of the journey was 5 miles?

(A) 5 miles

(B) 12 miles

(C) 25 miles

(D) 30 miles

(E) Cannot be determined by the information given.

This time around we only know that one of the legs is 5 miles long. We do not know which of the legs. Do you know another right triangle in which one each side is an integer and one of the sides equals 5? Yep, it’s our good friend the 3:4:5 triangle.

With this twist to the problem we no longer can say whether Michael drove in a 5:12:13 triangle or a 3:4:5 triangle, so the answer is (E).

In all likelihood, GRE would probably choose the first iteration of this problem, hoping to trap you with (E). The GMAT – an even more difficult test quant-wise – would be more likely to consider using the second iteration. But if you are reading this you are most likely taking the GRE – so remember, be wary of (E) cannot be determined.

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Hey Chris!!!

As far as I think, the answer for your first question should be (E) not (D) because:

(i) You have given only one information that the shortest side is 5, which doesn’t prove that it will follow 5-12-13 ratio unless the two shorter sides are ‘5’ and ’12’ then only the third side will be ’13’. It could be possible that sides would be 5-15-20 and still it will make right triangle.

(ii) If you wrote “what could be the total distance Michael drove if shortest side is 5”, then ’30’ can be the possible answer but you haven’t mentioned this in your question.

By what logic is 5-15-20 a right triangle?

Wow… this really lubricated my thought process …since im working, it seems my basic math skills were rusty after a long time off from studies… thank you so much 🙂

Thanks man!! I’m trying hard on maths section. Keep on posting.

You are welcome!

Hi,

Is not a leg one of the two shorter sides of the triangle? If so, the answer to the second question too will be 30 as in 3:4:5 triangle, 5 will not be a leg but the hypotenuse.

Please let me know if I am missing something here.

Thank you,

Ann

Hi Ann,

You definitely answered the question correctly, if I understand you correctly. There are two possible triangles (3:4:5 and a 5:12:13) and that’s why the answer is (E). Hope that makes sense :).

But the question clearly states that “if one of the legs was 5 miles”, which eliminates the possibility of 3:4:5 (since 5 in this case IS NOT leg but hypotenuse).

Hence, there is no ambiguity but the answer has to be 5:12:13.

Hope am clear this time.

Hi Ann,

That is definitely my fault :). I mixed up legs of the triangle with the hypotenuse. Honestly, I forget the proper use of the terminology ‘legs.’ Not that I think the GRE would write a question based on knowing the terminology.

So you are right. I will change the post by writing, “side of the triangle.”

Thanks for catching that :).

Actually, I changed that to “legs of the journey.” Which is hopefully unambiguous enough :).

🙂

Thanks for the tip. This might be one of the potential landmines I wont step on on the GRE now. 🙂

Great! Happy to have helped :).

Chirs:

Can you explain a bit how the ratio 3:4:5 and 5:12:13 are derived and In which scenario should we be using these?

Hi RM,

Unfortunately, that goes beyond the GRE, and my math skills :). At least I would really have to really think about it and in the process dust off those trigonometry books. Then again, it’s maybe just those weird things in nature that those numbers correspond to a right triangle. It definitely kept Pythagoras up at night.

Sorry couldn’t be more insightful :).

Hi Chris

The last piece of the solution is to crunch the numbers. How can we do it faster, without running many loops.

e.g.: In the later case, its easy to arrive at 3,4 given 5. Either by running fewer loops or by practice who knows the squares

In the later case of 12,13,5 its going to take more loops. we have 25. what next square addition makes another square. And we would start from say 2. Is there a smart way to keep the loops at minimal

Thanks a ton!

Siva

The key on a problem such as this one is to know the common special triangles. The GRE tests 3:4:5 and 5:12:13 over and over again. So as long as you are aware of these two triangles you should save yourself time and not feel you are “looping” looking for the correct answer.

Let me know if that helps!

Sure it does!

Thanks for the tips again!

in your first version of the question you have used the words “he drives z miles a straight line TOWARDS his starting point”. It does not say that he REACHED the starting pont. Hence we cant even be sure if its a triangle and then apply he Pythagorean triplet logic.

Oops, I do need to say that. Thanks for pointing it out :).

Hi Chris,

There is a Graphical representation for N E W S …. kinda arrow representation.

North is towards upwards and opposite to it shows south. Here comes my confusion where east direction is .? May this is a silly question .

Thanks.

E – is a right arrow, and W – is a left arrow.

Hope that helps :).

many thanks chris it was so helpful

You’re welcome 🙂

Hi Chris,

In case where D is the correct answer.

It can be 45-45-90 Triangle too. x may be equal to y/z?

Please comment.

Hi Shri,

Actually, the first question says that all x, y, and z are integers. Because of the root2

requirement, there is no 45-45-90 triangle in which each side is an integer.

Hope that helps 🙂